First, brown the butter: take your ½ cup of butter, cut it into slices, and place them into a skillet over medium heat. Melt the butter, stirring frequently. It will begin to sputter and crackle. Continue stirring continuously to prevent burning. After a while, the crackling will subside and butter will begin to foam. Keep stirring. Once you see brown flecks and the melted butter turns a rich amber color, immediately remove it from heat and pour the butter into a separate, heatproof bowl.
Let the butter cool completely until it turns into a solid. You can speed this up by putting the butter in the fridge, but make sure it does not get too cold. You want it to be soft and at room temp--similar to if you were baking with normal butter.
Once the butter is ready, preheat your oven to 350°F. Grease and line three, 6" round cake pans.
Prepare the dry ingredients by mixing the flour, baking powder, and salt together until combined. Prepare the wet ingredients by mixing the buttermilk and sour cream together until smooth. If the wet ingredients are cold, give them a quick zap in the microwave to bring them to room temperature. Set these both aside for now.
Add the browned butter, oil, granulated sugar, and brown sugar into the bowl of an electric mixer fitted with the paddle attachment. Cream the butter and sugars together for 2 minutes on medium-high speed to combine them. Then, add 2 tbsp of dulce de leche and beat until light and fluffy.
Scrape down the sides of your bowl, then on low speed, add the eggs and then the vanilla extract, one at a time, beating until fluffy and fully combined.
Grab your prepared dry and wet ingredients. On the lowest speed, alternate adding ⅓ of the flour, ½ of the buttermilk/sour cream, another ⅓ flour, remaining liquid, and then remaining flour. Beat until just combined and do not overmix.
Transfer the batter to your cake pans (use a measuring cup to divide evenly) and bake for 30-35 minutes, or until the cakes are set. Let the cakes cool for 15 minutes in the pan before transferring them to wire racks to cool completely.
Chill the the cakes in the fridge or freezer for at least 1 hour before assembling to make it easier. You can pause here and keep them in the fridge/freezer overnight as well.